Global Well-posedness and Scattering of Defocusing Energy subcritical Nonlinear Wave Equation in dimension 3 with radial data

Abstract

The paper deals with the defocusing case of the energy subcritical non-linear wave equation in $R^3$. We assume the initial data is in the space $\dot{H}^s \times \dot{H}^{s-1}$ and radial. If $s=1$, this is the energy space and the scattering results are known. In this paper we choose $15/16<s<1$ and assume that the nonlinearity equals $-|u|^p u$ with $p = 2/{3/2 -s}$, so that this problem is $\dot{H}^s \times \dot{H}^{s-1}$ critical. We will prove the global well-posedness and scattering of the solution under the additional assumption that the $\dot{H}^s \times \dot{H}^{s-1}$ norm of the solution is uniformly bounded for all time $t$ in the maximal lifespan of the solution.